The cluster value problem in spaces of continuous functions

Johnson, W.; Ortega Castillo, S.

doi:10.1090/S0002-9939-2014-12190-3

The cluster value problem in spaces of continuous functions
HTML articles powered by AMS MathViewer

by W. B. Johnson and S. Ortega Castillo

Proc. Amer. Math. Soc. 143 (2015), 1559-1568

DOI: https://doi.org/10.1090/S0002-9939-2014-12190-3

Published electronically: December 10, 2014

PDF | Request permission

Abstract:

We study the cluster value problem for certain Banach algebras of holomorphic functions defined on the unit ball of a complex Banach space $X$. The main results are for spaces of the form $X=C(K).$

References

Fernando Albiac and Nigel J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. MR 2192298
R. M. Aron, P. D. Berner, A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. math. France 106 (1978), pp. 3-24.
R. M. Aron, B. J. Cole, T. W. Gamelin, Spectra of algebras of analytic functions on a Banach space, J. reine angew. Math 415 (1991), pp. 51-93.
Richard M. Aron, Daniel Carando, T. W. Gamelin, Silvia Lassalle, and Manuel Maestre, Cluster values of analytic functions on a Banach space, Math. Ann. 353 (2012), no. 2, 293–303. MR 2915537, DOI 10.1007/s00208-011-0681-0
Robert Braun, Wilhelm Kaup, and Harald Upmeier, On the automorphisms of circular and Reinhardt domains in complex Banach spaces, Manuscripta Math. 25 (1978), no. 2, 97–133. MR 500878, DOI 10.1007/BF01168604
C. Bessaga and A. Pełczyński, Spaces of continuous functions. IV. On isomorphical classification of spaces of continuous functions, Studia Math. 19 (1960), 53–62. MR 113132, DOI 10.4064/sm-19-1-53-62
B. J. Cole, T. W. Gamelin, W. B. Johnson, Analytic Disks in Fibers over the Unit Ball of a Banach Space, Michigan Math. J. 39 (1992), pp. 551-569.
Seán Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. MR 1705327, DOI 10.1007/978-1-4471-0869-6
J. Duncan and S. A. R. Hosseiniun, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), no. 3-4, 309–325. MR 559675, DOI 10.1017/S0308210500017170
Theodore W. Gamelin, Analytic functions on Banach spaces, Complex potential theory (Montreal, PQ, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 439, Kluwer Acad. Publ., Dordrecht, 1994, pp. 187–233. MR 1332962
Jorge Mujica, Complex analysis in Banach spaces, North-Holland Mathematics Studies, vol. 120, North-Holland Publishing Co., Amsterdam, 1986. Holomorphic functions and domains of holomorphy in finite and infinite dimensions; Notas de Matemática [Mathematical Notes], 107. MR 842435
A. Pełczyński and Z. Semadeni, Spaces of continuous functions. III. Spaces $C(\Omega )$ for $\Omega$ without perfect subsets, Studia Math. 18 (1959), 211–222. MR 107806, DOI 10.4064/sm-18-2-211-222
Shôichirô Sakai, $C^*$-algebras and $W^*$-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60, Springer-Verlag, New York-Heidelberg, 1971. MR 0442701
Jean-Pierre Vigué, Le groupe des automorphismes analytiques d’un domaine borné d’un espace de Banach complexe. Application aux domaines bornés symétriques, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 2, 203–281. MR 430335